Why are solitons stable?

Abstract

Abstract. The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, solitary wave or soliton solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions can be relatively straightforward, but the fact that they are stable requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the critical case in which the nonlinearity is at exactly the right power to potentially allow for a self-similar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and Grillakis-Shatah-Strauss, to the more recent asymptotic stability and blowup analysis of Martel-Merle and Merle-Raphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations.

Stability of stationary solutions for nonintegrable peakon equations

“…These results are a first step towards understanding how the soliton resolution conjecture, as described in [40], should hold for (1); this would be consistent with the numerical results of Holm and Staley (see Figure 1). However, our notion of stability is rather limited, in that it requires solutions that are initially close to the lefton with respect to the Banach space norm · Z , in order to be close in H 1 at subsequent times.…”

“…The behaviour observed separately in each of the parameter ranges b > 1 and b < −1 can be understood as particular instances of the soliton resolution conjecture [40], a vaguely defined conjecture which states that for suitable dispersive wave equations, solutions with "generic" initial data will decompose into a finite number of solitary waves plus a radiation part which decays to zero. In this article, our aim is to provide a first step towards explaining this phenomenon analytically for the equation (1) in the "lefton" regime b < −1.…”

Interaction of Coherent States for Hartree Equations

“…In those papers, the initial amplitude a is very specific, since it is a ground state. The propagation and stability of multi-solitons for the nonlinear Schrödinger equation (without external potential) have been studied in [33][34][35]37] (see also [40]). In the framework of these papers, the waves do not interfere.…”

Uniform stabilization of numerical schemes for the critical generalized Korteweg-de Vries equation with damping

“…There is a large body of literature for the analysis of the GKdV equation on the line, the half-line, the torus, and boundary domains (see for example, [5,12,16,19,22,25,26,34]). The case p = 1 is essentially the KdV equation, and the case p = 2 is known as the modified Korteweg-de Vries (mKdV) equation.…”

“…The case p = 1 is essentially the KdV equation, and the case p = 2 is known as the modified Korteweg-de Vries (mKdV) equation. The case p = 4 is particularly interesting due to its mass-critical nature [22,25,26,34]. More precisely, local results for any initial data in L 2 established by Kenig et al [16], can be extended globally for the case p ≥ 4 only for small initial data.…”